PROFESSOR AT HEIDELBERG 181 



or from luck, discovers a new track that leads him on a little, 

 till at length when he reaches the summit he finds to his shame 

 that there is a royal way, by which he might have ascended, 

 had he only had the wits to find the right approach to it. In 

 my works I naturally said nothing about my mistakes to the 

 reader, but only described the made track by which he may 

 now reach the same heights without difficulty/ 



The Theory of Organ Pipes had till then been treated on 

 the assumption that the motion of the aerial particles within 

 the tubes was everywhere parallel to their axis, and that both 

 velocity and pressure were equal at all points of the same cross- 

 section of the tube, a view that was valid for the parts of a 

 cylindrical or prismatic tube more remote from the open ends, 

 but was inadmissible near the open ends where the waves 

 which are plane in the tube spread out from it in the form of 

 spherical waves ; for such a transition could not come about 

 suddenly. The view of Bernouilli, Euler, and Lagrange that 

 the condensation of the air at the open end of the tube was 

 nil was equally inaccurate, since the density there cannot be 

 taken as equal to that of the undisturbed air, but only to the 

 altered density of the adjacent air that is itself thrown into 

 vibration in the free space. Helmholtz followed up his earlier 

 work in acoustics by an exact theoretical inquiry into the 

 question as to the manner in which plane sound-waves, pro- 

 duced in the depth of a cylindrical tube, behave on their escape 

 into free space. He settled this very difficult problem mathe- 

 matically, without resort to hypothesis, by setting himself to 

 discover what form of vibration is permanently set up, when 

 the cause of the vibrations is allowed to act continuously and 

 uninterruptedly. In accordance with his earlier theory, he 

 assumed that the vibrations correspond with those of a simple 

 tone, since all complex vibrational forms can be considered as 

 due to the summation of a number of such simple tones. 



After applying the most important general laws of the Func- 

 tions of Electrical Potential to the Theory of Sound-waves, he 

 goes on to his particular problem of determining the motions 

 of air at the open end of a cylindrical tube, when plane waves, 

 corresponding to a simple tone, are produced within the tube 

 from any cause, and communicate their motion from the mouth 

 of the tube to the external air when it is affected by no other 



